Classroom Interactions
and Linguistic Capital: A Bourdieuian Analysis of the Construction of Social
Difference In Mathematics Education

Robyn Zevenbergen

Griffith University
- Gold Coast Campus

Goodenough (1976) asked
the question "What do people need to know in order to operate in a manner
that is acceptable to others in a society?" This question provides the
stimulus for this paper where the question is rephrased to: "What do students
need to know in order to operate in a manner which is acceptable in the
mathematics classroom?" Such a question is not without political implications
and so needs to be extended to include questions about the consequences
of participation in mathematics classroom. To answer this question, I appropriate
Gore's (1990) notion of "pedagogy as text" and develop the argument that
mathematics pedagogy is a text which students must be able to read in order
to be constructed as effective learners of mathematics. These texts, however,
are not apolitical and as Bourdieu (in (Wacquant, 1989) has argued persuasively,
language is a form of capital which can be exchanged for other forms of
capital - social, economic or cultural. Combining these two frameworks,
I argue that students enter the mathematics classroom from a range of socio-cultural
backgrounds whereby students whose socio-cultural background is congruous
with that of the culture represented in and through the practices embedded
within the mathematics classroom - including linguistic practices - are
more likely to be constructed as successful students.

Language as
a Form of Capital

Bourdieu (in Bourdieu
& Wacquant, 1992) argues that access to legitimate language, in this
case mathematics, is not equal and that linguistic competence is monopolised
by some. In considering the case of mathematics, this suggests that access
to the discourses and discursive practices of mathematics is differentially
accessible. For those students who enter the mathematics classroom with
a competence in the discursive practices, access to mathematics is made
more easily. Simultaneously, such students are more likely to be constructed
as successful students based on the teacher's judgement of their ability.
Within this context, language background is a form of capital which can
be converted to academic reward.

Linguistic competence
– or incompetence – reveals itself through daily interactions. Within the
mathematics classrooms, legitimate participation is acquired and achieved
through a competence in the classroom dialogic interactions. Students must
be able to display a discursive competence which incorporates a linguistic
competence, an interactional competence along with a discursive competence
if they are to be seen as competent learners of mathematics. Classroom
interactions are imbued with cultural components which facilitate or inhibit
access to the mathematical content. To gain access to this knowledge, students
must be able to render visible the cultural and political aspects of the
interactions.

Bourdieu (in Bourdieu
& Wacquant, 1992) argues that

Linguistic competence
is not a simple technical ability, but a statutory ability.

…what goes in verbal
communication, even the content of the message itself, remains unintelligible
as long as one doe not take into account the totality of the structure
of the power positions that is present, yet invisible, in the exchange.
(p. 146)

From their early years,
students are located within family structures and practices which will
facilitate the development and embodiment of particular cultural features,
least of which is language. For these students, the embodiment of their
cultural background into what Bourdieu refers to as the habitus, predisposes
them to think and act in particular ways. This embodiment of culture includes
a linguistic component. Students whose linguistic habitus is congruent
with that of the discursive practices represented in mathematics classrooms
are more likely to have greater access to the knowledge represented in
and through such practices.

From this perspective,
language must be understood as the linguistic component of a universe of
practices which are composted within a class habitus. Hence language should
be seen to be considered as another cultural product - in much the same
ways as patterns of consumption, housing, marriage and so forth. When considered
in this way, Bourdieu proposed that language is the expression of the class
habitus which is realised through the linguistic habitus.

Of all the cultural
obstacles, those which arise from the language spoke within the family
setting are unquestionably the most serious and insidious. For, especially
during the first years of school, comprehension and manipulation of language
are eth first points of teacher judgement. But the influence of a child's
original language setting never cease to operate. Richness and style of
expression are continually taken into account, whether implicitly or explicitly
and to different degrees. (Bourdieu, Passeron, & de saint Martin, 1994a)
p. 40To this end, the linguistic
habitus of the student will have substantial impact on his/her capacity
to make sense of the discursive practices of the mathematics classroom
and hence their subsequent capacity to gain access to legitimate mathematical
knowledge along with the power and status associated with that knowledge.
The processes through which the schooling procedures are able to value
one language and devalue others must be systematically understood. Through
this process, we will be better understand how mathematical pedagogy both
inculcates mathematical knowledge and imposes domination. In order to understand
how the linguistic practices of the mathematics classroom position and
hinder the effective participation of some students, the notion of pedagogy
as text is proposed.

Mathematics as a Text

Gore (1990) has argued
that pedagogy can be seen as a text which can be read and interpreted by
the reader. Texts can be read in a multiple of ways, so that the student
entering the mathematics classroom will be required to read, interpret
and make sense of what transpires in the mathematics classroom - not only
of the mathematical content, but also the pedagogical approaches within
which the content is relayed. To be able to read these texts, the students
must have some linguistic competence in the reading of social texts and
discursive practices.

The interactions that
occur within the classroom have been subjected to ethnomethodological approaches
and have been found to have highly ritualised components with clearly identifiable
discursive practices (Lemke, 1990; Mehan, 1982a). They argue that these
components are not explicitly taught but are embedded within the culture
of the classroom. The highly ritualised practices of classroom interactions
can be seen in the types of interactions which occur across the various
phases of the lesson. For example, the most common form of interaction
consists of a practice in which the teacher initiates a question, the students
respond and the teacher evaluates that response which Lemke (1990) refers
to as "triadic dialogue". This interactional practice can be observed in
the following:

T: What does area
mean?

S: The outside of
the square

T: Not quite, someone
else? Tom?

S: When cover the
whole surface, that's area.

T: That's good

Lemke (1990) argues that
this practice allows teachers to keep control of the content and flow of
this phase of the lesson. While Lemke focussed on the science classroom,
the style and purpose of this interaction can be just as readily applied
to the mathematics classroom and is aptly summed up as follows:Triadic dialogue
is an activity structure whose greatest virtue is that it gives the teachers
almost total control of the classroom dialogue and social interactions.
It leads to brief answers from students and lack of student initiative
in using scientific language. It is a form that is overused in most classrooms
because of a mistaken belief that it encourages maximum student participation.
The level of participation it achieves is illusory, high in quantity, low
in quality (Lemke, 1990, p. 168)This practice is not made
explicit to students, rather it must be learnt through implicit means.
To participate in the classroom interactions effectively, students must
have knowledge – either intuitive or explicit – of these unspoken rules
of interaction.

Furthering the work
of Lemke, Mehan (1982b) has identified three key phases of a lesson - the
introduction, the work phase and the concluding/revision phase. In each
of these phases there is a shift in the power relations between the students
and teacher which permits different forms of interactions to occur (Mehan,
1982b; Schultz, Florio, & Erickson, 1982). For the purposes of this
paper, it is my intention to discuss the introductory phase only.

Mehan (1982) argues
that during the intoductory phase of the lesson, the teacher maintains
tight control over the students, initially to ensure that the students
are ready for the content of the lesson. Once control has been established
and attention gained, the lesson can then proceed. Triadic dialogue is
commonly observed in this phase in order to keep control of the academic
content of the lesson and the control of the students. Dialogue between
students and between teacher and students is not generally part of this
phase. If the teacher initiates a question but the student is not able
to respond, it is not appropriate for students to express their lack of
understanding since this will interrupt the flow of the phase. If there
is a misunderstanding or lack of understanding, it is more appropriate
for this to be voiced in the work phase of the lesson.

What is lacking from
this corpus of knowledge of classroom interaction is the failure to recognise
that these interactions recognise a particular linguistic form which will
be more accessible to some students than others. In this sense, the interactions
within the classroom can be considered as another cultural product which
is more familiar to some students and not others. The linguistic habitus
of the students will or hinder a students capacity to render visible the
mathematical content embedded in the pedagogic action. The

…such cues [IRE]
are not necessarily "understood" by all participants, but they are certainly
part of the "functional conflict" between dominant and dominated languages
in (and out of) educational settings. (Collins, 1993, p.131)In the previous sections,
I have drawn on the work from a number of traditions – theoretical and
methodological – and have proposed that the social background of the student
will the construction of a particular linguistic habitus. The field of
mathematics, having its own regulatory discourses and discursive practices,
will recognise and value some linguistic practices and not others. These
practices are socially biased.

Method

An ethnography of two
classrooms was undertaken in which mathematics lessons were videotaped.
The two classrooms were located in socially divergent sites - one an independent
school which serves a middle- to upper-class clientele (Angahook). The
other classroom was in a state school serving a predominantly working-class
clientele (Connewarre). The classrooms were in the second last year of
primary school and most students were approximately 10-11 years old. The
video-taped lessons were transcribed and analysed. Extracts from one of
the lessons from each classrooms will be used as examples for this paper.

Angahook

This school serves
a middle- to upper-class client group. The mathematics teaching, learning,
assessment and curriculum are relatively conservative with a strong emphasis
on rote learning, preparation for examinations and teacher-directed pedagogy.
The class sizes are small with only 12 students in classroom observed.
In the lesson presented here, the students were undertaking an activity
from the Mathematics Curriculum Teaching Package. Prior to the extract
shown, the teacher (Helen) has used a number of short mental arithmetic
tasks. The following is the introduction to the lesson.

T: You are asked
to judge the diving for the Olympics, you will need to know the degree
of difficulty because what if someone did just a plain dive and did it
perfectly and got full marks for it and what if someone else did a triple
somersault, back flip, side swinger double pike and knocker banger and
only got half marks for ti because they entered the water and made a bit
of a splash. Is that fair?

C: No

T So we have to
talk about degrees of difficulty. What do you think that means? What does
that actually mean? Robert?

Robert You have
to add a bit more to the score because of the degrees of difficulty.

T Good boy. Yes,
good. Daniel?

Daniel Well the
performance of their dive, how they dive and well like they might have
a very good dive and make a very big splash and may even get off

T Right, good. OK
you are on the right track. What do you want to say about degree of difficulty
Cate?

Cate How hard it
is?

T How hard it is.
Tom what would you like to say about degree of difficulty? That's not a
word we use much in our everyday language..... degree of difficulty.

Tom The percentage
of how hard it is

T Good. Because
you're focussing on the word degree though aren't you. So a really hard
dive. Now you can see on this sheet they're talking about DD which is short
for degree of difficulty and a really hard dive. What would be a really
hard dive? What would be the highest number for a degree of difficulty
be? Have a look at your sheet. Try and work out the degree of difficulty.
Vicky?

Vicky 8

From this extract it can
be seen that the teacher follows the triadic dialogue identified by Lemke.
The teacher retains control of the content and interactions through the
use of the three phases of interactions. Using this approach she is able
to control the flow of the lesson as can be seen in the last interaction
where Tom has mentioned "percentages" which she then takes as a cue for
linking percentage and degree in a way which suits her purposes.

Examining the flow
of the interactions indicates that there is a complicit agreement between
the teacher and students to participate in the interactions. There are
no transgressions or challenges to the teacher's authority. This allows
for substantive content to be covered.

The teacher is able
to maintain control over both the form and content of the lesson and the
students through a mutual compliance with the implicit rules by both the
students and the teacher. She has used Triadic Dialogue to structure the
interactions and students infrequently transgress the rules. This allows
her to retain the focus of the lesson and in so doing, the students are
exposed to a significant amount of mathematical knowledge that is embedded
in that dialogue. The teacher's capacity to deliver the lesson in this
way allows for her to use a very rich mathematical language as she discusses
the mathematical content. In other words, the students are exposed to mathematical
language and concepts in a style which takes for granted their linguistic
background. The work of Brice-Heath (1982) has shown that middle-class
students are more likely to be familiar with these forms of school interactions
due to their similarity with the linguistic patterns of the home environment.
This familiarity has facilitated a linguistic habitus which is similar
to that of the formal mathematics classroom and hence permits access to
the codes and signifiers of school mathematics.

Connewarre

Connewarre is a large
government school which is located within a large housing commission estate.
The clientele of the school is predominantly working class with many of
the parents receiving government support. The classrooms are smaller than
Angahook with approx 25-30 students in each class. The teacher introduces
the mathematics lessons with problem solving activities which the students
undertake as small groups. They are able to be physically involved in the
activities and it is not uncommon for the students to draw on the carpet
with chalk to represent the task or physically construct the problem. The
mathematics has a strong emphasis on real life situations. The following
extract is the introduction to a lesson in which the teacher has drawn
a net on the board which the students will have to draw onto card and then
construct. Students are then required to develop a number of nets for nominated
prisms.

T So if I put
those together we start talking more about a shape I am talking about.
It's sort of a rectangle on the sides, all the way round but you don;t
call it is a rectangle, because a rectangle is jus the flat surface. What
do you call the whole thin if that was one whole solid shape. What do you
call that?

C A cube

T He said a cube.
Don't call out please.

C A rectangular
rectangle

T You're on the
right track

C A 3D rectangle

T Three dimensions,
technically I suppose you're right.

C A rectangular

It's a rectangular
something. Does anyone know what it is called?

C A parallelogram

T Put your hand
up please.

C [unclear]

T No

More calling out

T I guess you could
have a rectangular parallelogram, but no. A rectangle is a special parallelogram.,

C A rectangular
oblong

T The word we are
looking for is prism

C Yeah that's what
I said

T Say the word please

Cs Prism

T Not like you go
to jail "prison", that's prison. Excuse me, could you return those please.

[calling out]

T So one thing that
we think about with rectangular prisms and that this shape on here is,
excuse me...Now you can leave them down please. You need a little bit of
practice at lunch because you can;t stop fiddling. This shape here is drawn
out on the graph, this grid here [net for a rectangular prism]. We 're
going to try and do the same thing. Draw the shape and then cut it out.
If you look at the shape, it's made up of rectangles and squares.

In this extract, it indicates
that the flow of the lesson and content is hampered by the challenges to
the teacher's authority. The triadic dialogue does not serve the same purpose
as noted by Lemke (1990) and found at Angahook. The field of mathematics
education has particular unspoken rules of interaction which have not been
appropriated by the students at Connewarre, or may be resisted by the students.

There are many transgressions
of the implicit rules of classroom interactions. The flow of the lesson
is fragmented as students challenge the teacher's control for the floor
and content of the lesson.

The linguistic habitus
of the student implies a propensity to speak in particular ways which,
as can be observed in the case of the interactions in this extract, works
to exclude students from the mathematical content. The students are not
as competent in the linguistic exchanges of the mathematical interactions
as their middle-class peers thereby marginalising them in the process of
learning. The teaching of mathematics in this way tacitly presupposes that
the students will have the discursive knowledge and dispositions of particular
social groups, namely the middle-class. The students are not as complicit
in the classroom practices and in so doing are being excluded from active
and full participation in the mathematics of the interactions. In this
way, students have been exposed to the symbolic violence of formal education.

Conclusion

Using data from mathematics
classrooms, Voigt (1985, p. 81) has argued, "The hidden regularities, the
interaction patterns and routines allow the participants to behave in an
orderly fashion without having to keep up visible order" so the idea is
far from new. However, what I have sought to uncover using an interactionist
approach is the ways in which some students are able to gain access to
mathematical content and processes more readily than others. I have proposed
that one subtle and coercive way is through the linguistic habitus of the
students and the practices of classroom interactions whereby some students
enter the formal mathematics classrooms with a habitus that is akin to
that which is valorised within that context. These students will be able
to participate more effectively and efficiently than their peers for whom
the patterns of interaction are foreign to their habitus, thereby making
the habitus a form of capital which can be exchanged for academic success
within this context.

The predominantly implicit
codes of curriculum and classroom interactions take as a given that students
will have a familiarity with the legitimate linguistic practices of the
mathematics classroom, but neither curriculum nor pedagogy render that
language visible. gaining access to mathematical knowledge is facilitated,
or hindered, but a match or mismatch of codes. Rather than perceive this
a function of language deficiency, but as systemic through which the dominant
classes are able to maintain control:

pedagogies that tacitly
select the privileged and exclude the underprepared are not regrettable
lapses; they are systemic aspects of schooling systems serving class-divided
societies. (Collins, 1993, p.121)The linguistic habitus
of the middle-class students predisposes them so that their possession
of knowledge of what constitutes "appropriate" classroom linguistic exchanges
is similar to that which the system values thus allowing them to participate
in effective classroom practice. Alternatively, the linguistic habitus
facilitates the appropriation what the system offers. The dispositions,
as per the linguistic habitus, each of the classes have facilitate or hinder
their acquisition of mathematics. The linguistic habitus is differentially
valued within the mathematics classroom so that for some students the linguistic
code with which they are familiar and use within the classroom becomes
a form of capital which can be exchanged for other cultural goods - in
this case, grades and the subsequent academic success conveyed to the individual.
"The more distant the social group from scholastic language, the higher
the rate of scholastic mortality (Bourdieu, Passeron, & de saint Martin,
1994b, p.41).

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Applied Linguistics.

Schultz, J. J., Florio,
S. &., & Erickson, F. (1982). Where's the floor? Aspects of the
cultural organisation of social relationships in communication at home
and in school. In P. Gilmore & A. A. Glatthorn (Eds.), Children
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